The crossing-number conjecture for Sunlet–Star Cartesian products

Let $\mathcal{S}_n$ be the Sunlet graph on $2n$ vertices, obtained by attaching $n$ pendant edges to the cycle $C_n$, and let $K_{1,m}$ be the star graph on $m+1$ vertices. For a graph $G$, write $cr(G)$ for its crossing number. Sunlet–Star crossing-number conjecture. For $n\geq 3$ and $m\geq 1$,

$$cr(\mathcal{S}_n\Box K_{1,m})=n\frac{m(m-1)}{2}.$$

The formula is established in the paper for $m=1,2,3$ and is supported computationally for $n,m\leq 20$, but remains conjectural for general $m$.